D-Separation. b) the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C.

Transcription

1 D-Separation Say: A, B, and C are non-intersecting subsets of nodes in a directed graph. A path from A to B is blocked by C if it contains a node such that either a) the arrows on the path meet either head-to-tail or tail-totail at the node, and the node is in the set C, or b) the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C. If all paths from A to B are blocked, A is said to be d-separated from B by C. Notation: 1

2 D-Separation Say: A, B, and C are non-intersecting subsets of nodes in a directed graph. A path from A to B is blocked by C if it contains a node such that either a) the arrows on the and path meet not either of head-to-tail or tail-totail at the node, and the node is in the set C, or b) the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C. If all paths from A to B are blocked, A is said to be d-separated from B by C. Notation: D-Separation is a property of graphs probability distributions 2

3 D-Separation: Example We condition on a descendant of e, i.e. it does not block the path from a to b. We condition on a tail-to-tail node on the only path from a to b, i.e f blocks the path. 3

4 I-Map Definition 4.1: A graph G is called an I-map for a distribution p if every D-separation of G corresponds to a conditional independence relation satisfied by p: Example: The fully connected graph is an I-map for any distribution, as there are no D-separations in that graph. 4

5 D-Map Definition 4.2: A graph G is called an D-map for a distribution p if for every conditional independence relation satisfied by p there is a D-separation in G : Example: The graph without any edges is a D-map for any distribution, as all pairs of subsets of nodes are D-separated in that graph. 5

6 Perfect Map Definition 4.3: A graph G is called a perfect map for a distribution p if it is a D-map and an I-map of p. A perfect map uniquely defines a probability distribution. 6

7 The Markov Blanket Consider a distribution of a node xi conditioned on all other nodes: Markov blanket x i : all parents, children and co-parents of x i. at 7 Factors independent of x i cancel between numerator and denominator.

8 Repetition: Directed Graphical Models Directed graphical models can be used to represent probability distributions This is useful to do inference and to generate samples from the distribution efficiently 8

9 Repetition: D-Separation D-separation is a property of graphs that can be easily determined An I-map assigns every d-separation a c.i. rel A D-map assigns every c.i. rel a d-separation Every Bayes net determines a unique prob. dist. 9

10 In-depth: The Head-to-Head Node Example: a: Battery charged (0 or 1) b: Fuel tank full (0 or 1) p(a) =0.9 p(b) =0.9 a b p(c) c: Fuel gauge says full (0 or 1) We can compute and and obtain similarly: p( c b) =0.81 a explains c away p( c) =0.315 p( b c) p( b c, a)

11 Repetition: D-Separation 11

12 Directed vs. Undirected Graphs Using D-separation we can identify conditional independencies in directed graphical models, but: Is there a simpler, more intuitive way to express conditional independence in a graph? Can we find a representation for cases where an ordering of the random variables is inappropriate (e.g. the pixels in a camera image)? Yes, we can: by removing the directions of the edges we obtain an Undirected Graphical Model, also known as a Markov Random Field 12

13 Example: Camera Image x i x i directions are counter-intuitive for images Markov blanket is not just the direct neighbors when using a directed model 13

14 Markov Random Fields Markov Blanket All paths from A to B go through C, i.e. C blocks all paths. We only need to condition on the direct neighbors of x to get c.i., because these already block every path from x to any other node. 14

15 Factorization of MRFs Any two nodes x i and x j that are not connected in an MRF are conditionally independent given all other nodes: p(x i,x j x \{i,j} )=p(x i x \{i,j} )p(x j x \{i,j} ) In turn: each factor contains only nodes that are connected This motivates the consideration of cliques in the graph: Clique A clique is a fully connected subgraph. A maximal clique can not be extended with another node without loosing the property of full connectivity. Maximal Clique 15

16 Factorization of MRFs In general, a Markov Random Field is factorized as where C is the set of all (maximal) cliques and Φ C is a positive function of a given clique x C of nodes, called the clique potential. Z is called the partition function. Theorem (Hammersley/Clifford): Any undirected model with associated clique potentials Φ C is a perfect map for the probability distribution defined by Equation (4.1). As a conclusion, all probability distributions that can be factorized as in (4.1), can be represented as an MRF. 16

17 Converting Directed to Undirected Graphs (1) In this case: Z=1 17

18 Converting Directed to Undirected Graphs (2) x 1 x 3 x 1 x 3 x 2 x 2 x 4 x 4 p(x) =p(x 1 )p(x 2 )p(x 2 )p(x 4 x 1,x 2,x 3 ) In general: conditional distributions in the directed graph are mapped to cliques in the undirected graph However: the variables are not conditionally independent given the head-to-head node Therefore: Connect all parents of head-to-head nodes with each other (moralization) 18

19 Converting Directed to Undirected Graphs (2) x 1 x 3 x 1 x 3 x 2 x 2 x 4 x 4 p(x) =p(x 1 )p(x 2 )p(x 2 )p(x 4 x 1,x 2,x 3 ) p(x) = (x 1,x 2,x 3,x 4 ) Problem: This process can remove conditional independence relations (inefficient) Generally: There is no one-to-one mapping between the distributions represented by directed and by undirected graphs. 19

20 Representability As for DAGs, we can define an I-map, a D-map and a perfect map for MRFs. The set of all distributions for which a DAG exists that is a perfect map is different from that for MRFs. Distributions with a DAG as perfect map Distributions with an MRF as perfect map All distributions 20

21 Directed vs. Undirected Graphs Both distributions can not be represented in the other framework (directed/undirected) with all conditional independence relations. 21

22 Using Graphical Models We can use a graphical model to do inference: Some nodes in the graph are observed, for others we want to find the posterior distribution Also, computing the local marginal distribution p(x n ) at any node x n can be done using inference. Question: How can inference be done with a graphical model? We will see that, when exploiting conditional independences, we can do efficient inference. 22

23 Inference on a Chain The joint probability is given by The marginal at x 3 is In the general case with N nodes we have and 23

24 Inference on a Chain This would mean K N computations! A more efficient way is obtained by rearranging: Vectors of size K 24

25 Inference on a Chain In general, we have 25

26 Inference on a Chain The messages µ α and µ β can be computed recursively: Computation of µ α starts at the first node and computation of µ β starts at the last node. 26

27 Inference on a Chain The first values of µ α and µ β are: The partition function can be computed at any node: Overall, we have O(NK 2 ) operations to compute the marginal 27

28 Inference on a Chain To compute local marginals: Compute and store all forward messages,. Compute and store all backward messages, Compute Z once at a node x m : Compute for all variables required. 28

29 More General Graphs The message-passing algorithm can be extended to more general graphs: Undirected Tree Directed Tree Polytree It is then known as the sum-product algorithm. A special case of this is belief propagation. 29

30 Factor Graphs The Sum-product algorithm can be used to do inference on undirected and directed graphs. A representation that generalizes directed and undirected models is the factor graph. f(x 1,x 2,x 3 )=p(x 1 )p(x 2 )p(x 3 x 1,x 2 ) Directed graph Factor graph 30

31 Factor Graphs The Sum-product algorithm can be used to do inference on undirected and directed graphs. A representation that generalizes directed and undirected models is the factor graph. Undirected graph Factor graph 31

32 Factor Graphs Factor graphs can contain multiple factors x 1 x 3 f a for the same nodes f b are more general than undirected graphs x 4 are bipartite, i.e. they consist of two kinds of nodes and all edges connect nodes of different kind 32

33 Factor Graphs Directed trees convert to x 1 x 3 tree-structured factor graphs The same holds for x 4 undirected trees Also: directed polytrees convert to tree-structured factor graphs x 1 x 3 f a And: Local cycles in a directed graph can be removed by converting to a x 4 factor graph 33

34 Sum-Product Inference in General Graphical Models 1.Convert graph (directed or undirected) into a factor graph (there are no cycles) 2.If the goal is to marginalize at node x, then consider x as a root node 3.Initialize the recursion at the leaf nodes as: (var) or µ f!x (x) =1 µ x!f (x) =f(x) (fac) 4.Propagate messages from the leaves to x 5.Propagate messages from x to the leaves 6.Obtain marginals at every node by multiplying all incoming messages 34

35 Other Inference Algorithms Max-Sum algorithm: used to maximize the joint probability of all variables (no marginalization) Junction Tree algorithm: exact inference for general graphs (even with loops) Loopy belief propagation: approximate inference on general graphs (more efficient) Special kind of undirected GM: Conditional Random fields (e.g.: classification) 35

36 Conditional Random Fields Another kind of undirected graphical model is known as Conditional Random Field (CRF). CRFs are used for classification where labels are represented as discrete random variables y and features as continuous random variables x A CRF represents the conditional probability where w are parameters learned from training data. CRFs are discriminative and MRFs are generative 36

37 Conditional Random Fields Derivation of the formula for CRFs: In the training phase, we compute parameters w that maximize the posterior: where (x *,y * ) is the training data and p(w) is a Gaussian prior. In the inference phase we maximize 37

38 Conditional Random Fields Typical example: observed variables x i,j are intensity values of pixels in an image and hidden variables y i,j are object labels Note: the definition of x i,j and y i,j is different from the one in C.M. Bishop (pg.389)! 38

39 CRF Training We minimize the negative log-posterior: Computing the likelihood is intractable, as we have to compute the partition function for each w. We can approximate the likelihood using pseudo-likelihood: where Markov blanket C i : All cliques containing y i 39

40 Pseudo Likelihood 40

41 Pseudo Likelihood Pseudo-likelihood is computed only on the Markov blanket of y i and its corresp. feature nodes. 41

42 Potential Functions The only requirement for the potential functions is that they are positive. We achieve that with: where f is a compatibility function that is large if the labels y C fit well to the features x C. This is called the log-linear model. The function f can be, e.g. a local classifier 42

43 Training: CRF Training and Inference Using pseudo-likelihood, training is efficient. We have to minimize: Log-pseudo-likelihood Gaussian prior This is a convex function that can be minimized using gradient descent Inference: Only approximatively, e.g. using loopy belief propagation 43

44 Summary Undirected models (aka Markov random fields) provide an intuitive representation of conditional independence An MRF is defined as a factorization over clique potentials and normalized globally Directed and undirected models have different representative power (no simple containment ) Inference on undirected Markov chains is efficient using message passing Factor graphs are more general; exact inference can be done efficiently using sum-product 44